Define $$ 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} $$
$$ i = \begin{bmatrix} 0 & -1 \\ 1 & 0 \\ \end{bmatrix} $$
$$ j = \begin{bmatrix} 0 & -i \\ -i & 0 \\ \end{bmatrix} $$
$$ k = \begin{bmatrix} i & 0 \\ 0 & -i \\ \end{bmatrix} $$
Define
$$ H = a[1]+b[i]+c[j]+d[k]$$
$$ H = \begin{bmatrix} a+di & -b-ci \\ b-ci & a-di \\ \end{bmatrix} $$
$$ H = \begin{bmatrix} z & w \\ -\bar w & \bar z \\ \end{bmatrix} $$
where $$z= a+di$$ $$w=-b-ci$$
This is the basic quaternion setup we've been taught, my question is the following:
Define $\alpha$, $\beta$ are matrices in the form of $H$. What is $\alpha \cdot \beta$?
For example before we even get to multiplying the two H matrices, what does one H matrix represent in terms of a geometric interpretation? Is it a rotation in $R^4$? Am I even interpreting what the question is asking properly?